And there is the rub!
The two differing definitions of Active Speaker systems.
One is (yours) that any powered speaker is Active.
The other is that, it is active only if the crossover is active.
The notion of active vs passive, is determined by the crossover topology employed.
So a bi-amp ed, otherwise passive speaker is still a passive system.
I endorse the latter definition.
How could the latter definition possibly be correct? For instance, how could you possibly describe a single driver crossover less speaker with its's own inbuilt amplifier and only line level inputs as a passive loudspeaker?
Surely active simply means line level inputs and built in amplifiers, where and how the crossover is implemented isn't a defining characteristic of active speakers although, typically, an active speaker would have multiple drivers, inbuilt amplifiers and inbuilt, usually active, crossover.